Optimal. Leaf size=34 \[ \frac{a^2 x^4}{4}+\frac{6}{13} a b x^{13/3}+\frac{3}{14} b^2 x^{14/3} \]
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Rubi [A] time = 0.0291076, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{a^2 x^4}{4}+\frac{6}{13} a b x^{13/3}+\frac{3}{14} b^2 x^{14/3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt [3]{x}\right )^2 x^3 \, dx &=3 \operatorname{Subst}\left (\int x^{11} (a+b x)^2 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (a^2 x^{11}+2 a b x^{12}+b^2 x^{13}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{a^2 x^4}{4}+\frac{6}{13} a b x^{13/3}+\frac{3}{14} b^2 x^{14/3}\\ \end{align*}
Mathematica [A] time = 0.0231477, size = 34, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{6}{13} a b x^{13/3}+\frac{3}{14} b^2 x^{14/3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{6\,ab}{13}{x}^{{\frac{13}{3}}}}+{\frac{3\,{b}^{2}}{14}{x}^{{\frac{14}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.985159, size = 270, normalized size = 7.94 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{14}}{14 \, b^{12}} - \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13} a}{13 \, b^{12}} + \frac{55 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a^{2}}{4 \, b^{12}} - \frac{45 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{3}}{b^{12}} + \frac{99 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a^{4}}{b^{12}} - \frac{154 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{5}}{b^{12}} + \frac{693 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{6}}{4 \, b^{12}} - \frac{990 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{7}}{7 \, b^{12}} + \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{8}}{2 \, b^{12}} - \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{9}}{b^{12}} + \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{10}}{4 \, b^{12}} - \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{11}}{b^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47228, size = 72, normalized size = 2.12 \begin{align*} \frac{3}{14} \, b^{2} x^{\frac{14}{3}} + \frac{6}{13} \, a b x^{\frac{13}{3}} + \frac{1}{4} \, a^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.09438, size = 31, normalized size = 0.91 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{6 a b x^{\frac{13}{3}}}{13} + \frac{3 b^{2} x^{\frac{14}{3}}}{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11541, size = 32, normalized size = 0.94 \begin{align*} \frac{3}{14} \, b^{2} x^{\frac{14}{3}} + \frac{6}{13} \, a b x^{\frac{13}{3}} + \frac{1}{4} \, a^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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